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2 votes
0 answers
491 views

Is there a Bayesian theory of deterministic signal? Prequel and motivation for my previous question

This is a prequel to my question: What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory) Clearly my ...
Fabrice Pautot's user avatar
7 votes
1 answer
621 views

Does every (generalized?) Markov chain admit transition probabilities?

To pose the question let us start by recalling the following notions: Transition Probabilities. A transition probability matrix between two measurable spaces $(S,\mathcal{S})$ and $(V,\mathcal{V})$ ...
David's user avatar
  • 486
6 votes
2 answers
3k views

What's the probability distribution of a deterministic signal or how to marginalize dynamical systems? (functional integrals in probability theory)

Because I still have no idea how it is possible for me to write down seemingly important equations ... that don't make any sense (at least for me) and because I haven't got any helpful comment so far, ...
Fabrice Pautot's user avatar
6 votes
1 answer
340 views

Law of unconsious statistician: application in characteristic function

Let $g(x)=(x-a)\mathbf 1_{x\ge a}$ for some $a>0$ and let $X$ be a non-negative random variable with cdf $F$ and $E[X]<+\infty$. I want to calculate $$\frac{d}{da}E[g(X)]$$ To do that I thought ...
Jimmy R.'s user avatar
1 vote
0 answers
302 views

Integration involving modified bessel function, exponential and power

I need to find the following integration. $$ \int_0^a e^{-(N-1)x}(\sqrt{4x}K_{1}(\sqrt{4x}))^N $$ where $$ a>0, \quad N \geq 1 $$ Any help will be much appreciated. BR Frank
Frank Moses's user avatar
0 votes
0 answers
453 views

Integral involving modified bessel function of second kind, exponential and power

I need to compute the following integral. $$ \int_0^a e^{-bx}\sqrt{4(a-x)}K_1(\sqrt{4(a-x))}dx\,. $$ where $$ a>0$$ and $b$ can be greater than zero or less than zero but it is not a complex ...
Frank Moses's user avatar
0 votes
0 answers
81 views

Why is $\mathcal{E}(X)=\mathcal{E}(X,X^*)$?

According to a course about $\sigma$-agebras in infinite dimensional space they said that it is easy to see that : $$\mathcal{E}(X)=\mathcal{E}(X,X^*)$$ where: $X$ is separable real Banach space. $\...
Heidy's user avatar
  • 121
2 votes
1 answer
446 views

Is the following "section-wise" defined function measurable in the product space?

I asked this question in mathstackexchange a couple of days ago. Almost right after posing it a partial (affirmative) answer came to my mind in the following form Proposition: Assume that $(X,\...
David's user avatar
  • 486
3 votes
0 answers
286 views

Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go: Each bidder's valuation $\theta _{i},$ $i=1,...,N$, is ...
Econ's user avatar
  • 31
6 votes
2 answers
3k views

Weak convergence of random measures

Let $\mu_n,n\in \mathbb N$ be a random probability measures and let $\mu$ be a deterministic probability measure on $\mathbb R$. That is to say, that the $\mu_n$ are measurable maps from a probability ...
whz's user avatar
  • 255
0 votes
0 answers
454 views

Reference: Bochner Integral`

What would be an easily accessible book dealing with Bochner integration as applied to probability theory (I'm looking to understand random elements and their basic related concepts in a formal yet ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
232 views

Morphisms associated to measured spaces [duplicate]

In a previous discussion (von neumann algebras and measurable spaces), the connexion between von Neumann algebras and localized measured spaces was clarified. I would like to have a category theory ...
Issam Ibnouhsein's user avatar
0 votes
1 answer
220 views

Behavior of the integral of products of probability densities

Assume $z \in \mathbb{R}^m$ and $x \in \mathbb{R}^n$. Assume we have proper density function $P(z)$ and proper conditional density function $P(x|z)$. We give the definition $$ T(x_1,\ldots,x_n) := \...
Yicong Liang's user avatar
4 votes
1 answer
268 views

An inequality concerning convexity and expectation

Assume $f$ and $g$ are nonnegative with $$\int_0^\infty f(x)dx=1=\int_0^\infty g(x)dx $$ and $$\int_0^\infty xf(x)dx<\infty > \int_0^\infty xg(x)dx $$ Is it true for nonnegative numbers $p$, $q$ ...
Bjørn Kjos-Hanssen's user avatar
1 vote
1 answer
278 views

Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $F$ is the Gauss' hypergeometric function

What is the Laplace transform of : $t^{\gamma-1} F(\alpha,\beta,\delta,t)$, where $\gamma >0 $ and $F$ is the Gauss' hypergeometric function. Thanks!
tam's user avatar
  • 233
1 vote
0 answers
251 views

Inflated independent samples for Monte Carlo estimation

In my particular problem, running an MCMC is too expensive, so I'm looking for a simple MC estimator, which would partially inherit the correlated samples of MCMC, yet would not require computing ...
Anton's user avatar
  • 101
2 votes
2 answers
407 views

How to calculate $P(\sum_{i=1}^{m}(A_i+S_i)\le L)$ with $A_i,L\sim\text{exp}(\lambda),S_i\sim\text{exp}(\mu)$ and positive integers $\lambda\neq\mu$?

Recently I was stumped by the calculation of the probability $$\mathbb{P} \big(\sum_{i=1}^{m} (A_i + S_i) \le L < \sum_{i=1}^{m+1} (A_i + S_i) \big)$$ where $A_i \sim \text{exp}(\lambda), S_i \sim ...
hengxin's user avatar
  • 139
4 votes
0 answers
428 views

Inverse of matrix-valued function

Given $c>0$. Let $\gamma_c:{\cal M}_{k \times k}^+\mapsto {\cal M}_{k \times k}^+$ is a function defined by \begin{equation} \gamma_c(\Omega)=\frac1{\sqrt{(2\pi)^{k}|\Omega|}}\int_{\mathbb{R}^k}\{(-...
Jlamprong's user avatar
  • 133
1 vote
1 answer
142 views

Characterization of a particular integrable function

Let $f$ be a strictly positive function such that $\int_{0}^{\infty}f(x)dx=\int_{0}^{\infty}xf(x)=1$ (i.e., a probability density function with expectation one). Let also $g$ be a nonnegative ...
Gili's user avatar
  • 13
4 votes
1 answer
740 views

Integral wrt probability measure

Let $\Theta\subseteq\mathbb{R}^d$ is open set and $(\cal X, \cal A)$ be a measurable space . For every $\theta\in\Theta$, suppose that $P_\theta$ is a probability measure on $(\cal X, \cal A)$. ...
Jlamprong's user avatar
  • 133
2 votes
1 answer
563 views

Prove or disprove $ \int_0^\infty \int_{-x}^0 f(x)f(y)\,dy\,dx > \int_0^\infty \int_{-\infty}^{-x} f(x)f(y)\,dy\,dx. $

Consider a symmetric, unimodal distribution $f(x)$ such that $\int_0^\infty f(x)\,dx > 1/2$. I want to prove or disprove the following: $$ \int_0^\infty \int_{-x}^0 f(x)f(y)\,dy\,dx > \int_0^\...
ubaabd's user avatar
  • 175
3 votes
1 answer
2k views

From Lebesgue Integral to Stieltjes Integral, and integration by parts

Let $X$ be a real random variable with c.d.f function $F$. Let $g$ be an increasing measurable real function and assume that $\mathbb{E}\left[g(X)\right]$ exists (and is finite). What additional ...
Adrien's user avatar
  • 591
1 vote
0 answers
100 views

Distribute Monte Carlo samples among dimensions

Simplified problem: Given a $d$-times nested convolution of an input function $g(x):\mathbb{R}\mapsto \mathbb{R}$ with the same band-limited smooth function $f(x):\mathbb{R}\mapsto \mathbb{R}$. I am ...
Anton's user avatar
  • 101
2 votes
0 answers
341 views

Marginalizing multivariate normal over defined interval

Hello everyone, I am trying to obtain an analytic expression for the following Gaussian integral $$\frac{1}{\sqrt{(2 \pi)^n |\Sigma|}} \int \kern-0.2em \cdots \kern-0.2em \int d\mathbf{x}_{\sim i} \;...
amanoel's user avatar
  • 21
4 votes
1 answer
213 views

Practical way to check for geometric convergence

Target distribution is multimodal, 24 dimensions, continuous state space. For MCMC integration (MH sampler) I use a manually tuned proposal distribution. When I measure the convergence rate ...
Anton's user avatar
  • 101
9 votes
2 answers
646 views

Rain droplets falling on a table

Suppose you have a circular table of radius $R$. This table has been left outside, and it begins to rain at a constant rate of one droplet per second. The drops, which can be considered points as they ...
Nicolas Kim's user avatar
3 votes
2 answers
1k views

Is there a corresponding Hahn decomposition theorem for the real-valued Radon measures?

Hello, As we know that a signed measure $\mu$ on $R$ can be decomposed to the positive part $\mu_+$ and negative one $\mu_-$ by the Hahn decomposition theorem. My question is whether each real-...
Anand's user avatar
  • 1,649
3 votes
2 answers
994 views

measurability of integrated functions

DISCLAIMER: I'm not a mathematician, but a computer scientist, so I hope the question is not trivial (or perhaps I hope so, in order to get a definitive answer). Anyway it's not a homework, as ...
user avatar
4 votes
1 answer
562 views

Time-integral of a smooth, vector-valued function of a planar Brownian bridge

I'm looking for information on how to compute the distribution of the random vector $$Z = \int_0^t f(B_s) ds$$ where $t>0$ is fixed, $B_s$ is a 2D Brownian bridge with $B_0 = 0$, $B_t=b \in \...
Gabriel's user avatar
  • 43
7 votes
3 answers
966 views

Expectation of a simple function of multivariate gaussians iid rvs

I would like to compute analytically the following expected value: $$ E\left( \frac{X_i^2}{\sum_j \lambda_j^2 X_j^2}\right) $$ where the $X_i \approx N(0,1)$ are iid. It seems to be an elementary ...
gappy3000's user avatar
  • 461
0 votes
1 answer
551 views

Surface of the cut of an ellipsoid / Marginal density of a multivariate normal over an affine space

So I'm trying to get the marginal density of a multivariate normal over an affine space if $A$ is a matrix in $\mathbb{R}^p \times \mathbb{R}^n$ for $p < n$ and $B \in \mathbb{R}^n$, $\Sigma$ is a ...
Arthur B's user avatar
  • 1,902
5 votes
3 answers
945 views

An Integral and derived double integral

Suppose that $f\left(x\right)\geq0$ is continuous on $\left[-\infty,\infty\right]$ and $\int_{-\infty}^{\infty}f\left(x\right)dx=1$. Is it true that $\int_{-\infty}^{\infty}\left|x\right|f\left(x\...
user4606's user avatar
9 votes
1 answer
2k views

Kullback-Leibler divergence of scaled non-central Student's T distribution

What is the Kullback-Leibler divergence of two Student's T distributions that have been shifted and scaled? That is, $\textrm{D}_{\textrm{KL}}(k_aA + t_a; k_bB + t_b)$ where $A$ and $B$ are Student's ...
Neil's user avatar
  • 598

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